scispace - formally typeset
Search or ask a question

Showing papers by "Herbert Edelsbrunner published in 2006"


Proceedings ArticleDOI
05 Jun 2006
TL;DR: The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering and uses the algorithm to compute 1-parameter families of diagrams which are applied to the study of protein folding trajectories.
Abstract: Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithm's analysis is an elementary proof of the stability of persistence diagrams [7] in the special case of piecewise-linear functions. We use the algorithm to compute 1-parameter families of diagrams which we apply to the study of protein folding trajectories.

262 citations


Journal ArticleDOI
TL;DR: An algorithm for finding points of locally maximum elevation, which is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation.
Abstract: Given a smoothly embedded 2-manifold in ${\Bbb R}^3,$ we define the elevation of a point as the height difference to a canonically defined second point on the same manifold. Our definition is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation. We give an algorithm for finding points of locally maximum elevation, which we suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.

119 citations


Proceedings ArticleDOI
05 Jun 2006
TL;DR: It is proved that for functions f on a 2-manifold such ε-simplification exists, and an algorithm to construct them in the piecewise linear case is given.
Abstract: We continue the study of topological persistence [5] by investigating the problem of simplifying a function f in a way that removes topological noise as determined by its persistence diagram [2]. To state our results, we call a function g an e-simplification of another function f if ¦¦f−g¦¦∞≤e, and the persistence diagrams of g are the same as those of f except all points within L1-distance at most e from the diagonal have been removed. We prove that for functions f on a 2-manifold such e-simplification exists, and we give an algorithm to construct them in the piecewise linear case.

91 citations


Journal ArticleDOI
TL;DR: A definition of an interface surface formed by two or more proteins as a subset of their Voronoi diagram is presented and a hierarchy that distinguishes core and peripheral regions is defined, shown to have correlation with hot-spots in protein-protein interactions.
Abstract: Protein-protein interactions, which form the basis for most cellular processes, result in the formation of protein interfaces. Believing that the local shape of proteins is crucial, we take a geometric approach and present a definition of an interface surface formed by two or more proteins as a subset of their Voronoi diagram. The definition deals with the difficult and important problem of specifying interface boundaries by invoking methods used in the alpha shape representation of molecules, the discrete flow on Delaunay simplices to define pockets and reconstruct surfaces, and the assessment of the importance of topological features. We present an algorithm to construct the surface and define a hierarchy that distinguishes core and peripheral regions. This hierarchy is shown to have correlation with hot-spots in protein-protein interactions. Finally, we study the geometric and topological properties of interface surfaces and show their high degree of contortion.

44 citations


Patent
23 Jun 2006
TL;DR: In this paper, a feature skeleton is generated by partitioning three-dimensional object data into regions of a Morse complex, and then a plurality of smooth edges are replaced with corresponding pairs of curves that locate longitudinal boundaries of transitions between primary regions of the feature skeleton.
Abstract: Methods of modeling a three-dimensional surface structure include partitioning three-dimensional object data into regions of a Morse complex and generating a feature skeleton having a plurality of smooth edges and a plurality of vertices separating the regions of the Morse complex. Operations are also performed to thicken the feature skeleton by replacing the plurality of smooth edges with corresponding pairs of curves that locate longitudinal boundaries of transitions between primary regions of the feature skeleton. The thickening operations may also include replacing each of the plurality of vertices with a corresponding loop of edges, using setback-type vertex blends.

33 citations


Proceedings Article
01 Jan 2006
TL;DR: In this article, the problem of simplifying a function f in a way that removes topological noise as determined by its persistence diagram is investigated. But it is not shown that such simplifications exist for functions f on a 2-manifold.
Abstract: We continue the study of topological persistence [5] by investigating the problem of simplifying a function f in a way that removes topological noise as determined by its persistence diagram [2]. To state our results, we call a function g an -simplification of another function f if ‖f − g‖∞ ≤ , and the persistence diagrams of g are the same as those of f except all points within L1-distance at most from the diagonal have been removed. We prove that for functions f on a 2-manifold such -simplification exists, and we give an algorithm to construct them in the piecewise linear case.

23 citations